Section: New Results

Computability and Complexity

• Complexity of stream functions and higher-order complexity. We have pursued our works on higher-order complexity and the complexity of stream functions. Both notions are closely related as any function from natural numbers to natural numbers can be seen as a stream (an infinite list) of natural numbers:

  • A characterization of the class of Basic Feasible Functionals using term rewrite systems on streams and interpretation methods has been proposed in [13] . This result is part of Hugo Férée's PhD thesis for which he has obtained the Ackermann award.

  • In [14] , we have provided some interpretation criteria useful to ensure two kinds of stream properties: space upper bounds and input/output upper bounds. Our space upper bounds criterion ensures global and local upper bounds on the size of each output stream element expressed in term of the maximal size of the input stream elements. The input/output upper bounds criterion considers instead the relations between the number of elements read from the input stream and the number of elements produced on the output stream.

  • The paper [21] has extended the light affine lambda calculus with inductive and coinductive data types using the category theory notions of (weak) initial algebra and coalgebra.

• Complexity analysis of Object-Oriented programs. We have proposed a type system based on non-interference and data ramification (tiering) principles in [22] . We have captured the set of functions computable in polynomial time on OO programs. The studied language is general enough to capture most OO constructs and the characterization is quite expressive as it allows the analysis of a combination of imperative loops and of data ramification scheme based on Bellantoni and Cook’s safe recursion using function algebra.

• Rice-like theorem for primitive recursive functions. We have studied the following question: what are the properties of primitive recursive functions that are decidable (by a Turing machine), given a primitive recursive presentation of the function. We give a complete characterization of these properties. We show that they can be expressed as unions of elementary properties of being compressible. If $h:\mathbb{N}\to \mathbb{N}$ is a computable increasing unbounded function (like $log\left(n\right)$ or $2^n$), we say that a function $f:\mathbb{N}\to \mathbb{N}$ is $h$-compressible if for each $n$ there is a primitive recursive program of size at most $h\left(n\right)$ computing a function that coincides with $f$ on entries $0,1,...,n$. Whether $f$ is $h$-compressible is decidable given a primitive recursive program for $f$, and every decidable property can be obtained as a combination of such elementary properties. This result actually holds for any class of total functions that admits a sound and complete programming language. An article is currently in preparation.

• Parametrization of geometric figures. During the master internship of Diego Nava Saucedo, we have studied the semi-computability of geometric figures. A figure is semi-computable if there is a program that semi-decides whether a pixel intersects the figure. Our goal is to understand the semi-computability of a figure in terms of the parameters describing the figure. It turns out that the usual ways of parameterizing simple figures such as triangles, squares or disks do not behave well in terms of semi-computability. We have actually proved that no finite parametrization behaves well.

• Symbolic Dynamics on Groups. In an effort to better understand the interplay of geometry and computability in tiling theory, E. Jeandel has studied tiling problems on general Cayley graphs, and has obtained a significant number of new results. He has proven that groups with an (strongly) aperiodic tiling system have decidable word problem [30] , and provided examples of new groups (in particular monster groups) with such tiling systems, and proved that all nontrivial nilpotent groups have an aperiodic tiling system and an undecidable domino problem [31] . He also showed how the new concept of translation-like actions from geometric group theory can be used to prove that many groups, in particular the Grigorchuk groups and most groups with a nontrivial center, have an undecidable domino problem [33] .

• The smallest aperiodic tileset. In a joint work with Michael Rao, E. Jeandel has proven that there exists an aperiocic set of 11 Wang tiles [34] , and furthermore that this number is optimal.